Adding and Subtracting Polynomials

➕ Adding & Subtracting Polynomials

Part 1 of 5 — The Language of Polynomials

Topics in This Part

🔑 Key Concept: Adding and subtracting polynomials is really just combining like terms. Before we can combine them, we need to recognize them — that's what Part 1 is all about.

Terms, Coefficients, and Degree

A polynomial is a sum of terms. Each term is a number times a variable raised to a whole-number power.

In the term $7x^3$:

• The coefficient is $7$ (the number multiplied by the variable).
• The variable is $x$.
• The degree of the term is $3$ (the exponent on the variable).

The degree of the whole polynomial is the largest exponent that appears.

💡 A plain number like $8$ is called a constant term. Its degree is $0$ because $x^0 = 1$.

Find the Degree 🧮

Enter the degree of each polynomial (the largest exponent that appears).

1) $7x^4 - 2x^2 + 9$ 2) $3x + 5$ 3) $x^6$

Naming Polynomials

We name polynomials two ways — by their number of terms and by their degree.

By number of terms:

By degree:

🔑 So $x^2 + 3x - 4$ is a quadratic trinomial — quadratic (degree $2$) and trinomial (three terms).

Concept Check 🎯

What "Like Terms" Means

Like terms have the exact same variable part — the same variable raised to the same power. Only the coefficients can differ.

To combine like terms, add (or subtract) the coefficients and keep the variable part unchanged:

$3x^2 + 5x^2 = 8x^2 \qquad 4x - 9x = -5x$

⚠️ You can never combine $x^2$ with $x$. They are different "kinds" of term — like trying to add apples and oranges.

Like or Unlike? 🔽

For each pair, choose whether the terms are like terms.

➕ Adding & Subtracting Polynomials

Part 2 of 5 — Adding Polynomials

🔑 The Idea: To add two polynomials, drop the parentheses and combine like terms. Nothing else changes — the plus sign in front of a group does not flip any signs.

Adding Horizontally

Example: $(3x^2 + 5x + 2) + (4x^2 - 2x + 7)$

Step 1 — Remove the parentheses. Since both groups are added, every sign stays the same:

$3x^2 + 5x + 2 + 4x^2 - 2x + 7$

Step 2 — Group like terms (it helps to line up the same powers):

$(3x^2 + 4x^2) + (5x - 2x) + (2 + 7)$

Step 3 — Combine:

$7x^2 + 3x + 9$

💡 Notice the answer is written in standard form — terms in order from the highest power to the lowest. Always finish there.

Group the Like Terms 🔽

You are adding $(3x^2 + 5x + 2) + (4x^2 - 2x + 7)$. Choose the combined result for each group of like terms.

Adding Vertically

Some students prefer to stack the polynomials and line up like terms in columns, just like adding numbers:

Each column holds only like terms, so you just add down each column.

⚠️ Watch the gaps. If one polynomial is missing a power — say it has no $x$-term — leave that column blank (or write $0x$) so nothing lines up wrong.

Concept Check 🎯

A Fact Worth Knowing

When you add two polynomials, the result is always another polynomial. The degree of the sum is at most the larger of the two degrees — and sometimes it is less, when the leading terms cancel.

Example: $(x^2 + 3x) + (-x^2 + 5)$. The $x^2$ terms cancel:

$x^2 - x^2 + 3x + 5 = 3x + 5$

A sum of two quadratics became linear. Adding never raises the degree — it can only keep it the same or lower it. We'll use this idea again when we subtract.

Add the Polynomials 🧮

Add each pair and enter the coefficients of the result, in order.

1) $(4x^2 + 2x + 1) + (3x^2 + 6x + 5)$   Enter the coefficient of $x^2$, then $x$, then the constant.

2) $(7x + 9) + (2x - 4)$   Enter the coefficient of $x$, then the constant.

➕ Adding & Subtracting Polynomials

Part 3 of 5 — Subtracting Polynomials

🔑 The Big Rule: To subtract, distribute the minus sign to every term in the second polynomial — then add. The minus sign flips the sign of each term it reaches.

The Minus Sign Distributes

Subtracting a group means multiplying that whole group by $-1$:

$-(4x^2 - 2x + 7) = -4x^2 + 2x - 7$

Every sign flips: $+4x^2 \to -4x^2$, $\;-2x \to +2x$, $\;+7 \to -7$.

Example: $(3x^2 + 5x + 2) - (4x^2 - 2x + 7)$

Step 1 — Distribute the minus into the second group:

$3x^2 + 5x + 2 - 4x^2 + 2x - 7$

Step 2 — Combine like terms:

$(3x^2 - 4x^2) + (5x + 2x) + (2 - 7)$

Step 3 — Simplify:

$-x^2 + 7x - 5$

⚠️ The #1 mistake is flipping only the first sign. The minus hits all three terms — every single one.

Distribute the Minus 🔽

Rewrite $-(2x^2 - 6x + 4)$ by flipping each sign. Choose the correct sign for each term.

Another Worked Example

Example: $(6x^2 + x - 3) - (2x^2 + 4x - 9)$

Distribute the minus over the second group, then combine:

$6x^2 + x - 3 - 2x^2 - 4x + 9$

$(6x^2 - 2x^2) + (x - 4x) + (-3 + 9)$

$4x^2 - 3x + 6$

✅ Check by adding the opposite: subtracting $(2x^2 + 4x - 9)$ is the same as adding $(-2x^2 - 4x + 9)$. Lining those up gives $4x^2 - 3x + 6$ — same answer. ✓

Concept Check 🎯

Two Traps to Avoid

Before you practice, here are the two errors that trip up almost everyone:

⚠️ When the term you subtract is already negative, two minuses make a plus: $-(-1) = +1$. Slow down on those.

Subtract the Polynomials 🧮

Subtract and enter the coefficients of the result, in order.

1) $(8x^2 + 5x + 10) - (3x^2 + 9x + 2)$   Enter the coefficient of $x^2$, then $x$, then the constant.

2) $(2x - 7) - (6x - 1)$   Enter the coefficient of $x$, then the constant.

➕ Adding & Subtracting Polynomials

Part 4 of 5 — Word Problems & Applications

🔑 Why it matters: Polynomials model real quantities — perimeters, areas, costs, and profits. Combining them lets you build a single tidy expression for a whole situation.

Perimeter: Adding All the Sides

A triangle has sides of length $(2x + 1)$, $(3x - 4)$, and $(x + 6)$. Its perimeter is the sum of the sides:

$P = (2x + 1) + (3x - 4) + (x + 6)$

Drop the parentheses (all added) and combine like terms:

$P = (2x + 3x + x) + (1 - 4 + 6) = 6x + 3$

So the perimeter is $6x + 3$. If, say, $x = 5$, the perimeter is $6(5) + 3 = 33$ units.

💡 Adding polynomials lets one expression, $6x + 3$, stand in for every triangle in this family — just plug in the value of $x$ you need.

Find the Perimeter 🧮

A rectangle has length $(5x + 2)$ and width $(x - 3)$. Recall that a rectangle's perimeter is $P = 2(\text{length}) + 2(\text{width})$, which equals (length + width) added twice.

A quicker route: add the four sides $(5x+2) + (x-3) + (5x+2) + (x-3)$.

Enter the coefficient of $x$, then the constant, in the simplified perimeter.

"How Much More?" Means Subtract

Subtraction shows up whenever a problem asks for a difference — how much longer, how much more, how much is left over.

Example: Comparing two lengths

Board A is $(7x + 5)$ inches long. Board B is $(3x + 9)$ inches long. How much longer is Board A than Board B?

Difference $=$ (longer) $-$ (shorter):

$(7x + 5) - (3x + 9) = 7x + 5 - 3x - 9 = 4x - 4$

Board A is $(4x - 4)$ inches longer. (For $x > 1$ this is positive, as expected.)

⚠️ Order matters in subtraction. "How much longer is A than B" is $A - B$, not $B - A$.

Profit = Revenue − Cost

In business, profit is revenue minus cost. If both are modeled by polynomials, you subtract them.

Example

A small company's revenue (in dollars) for selling $x$ items is $R = 12x^2 + 40x$, and its cost is $C = 12x^2 + 15x + 200$.

$P = R - C = (12x^2 + 40x) - (12x^2 + 15x + 200)$

Distribute the minus and combine:

$P = 12x^2 + 40x - 12x^2 - 15x - 200 = 25x - 200$

The $12x^2$ terms cancel, leaving a clean linear profit model, $P = 25x - 200$. Selling $x = 8$ items gives $P = 25(8) - 200 = 0$, i.e. $0 — the break-even point.

Concept Check 🎯

Keyword Guide: Add or Subtract?

The hardest part of a word problem is deciding which operation to use. Watch for these signal words:

💡 When in doubt, ask: "Am I putting quantities together, or finding how far apart two quantities are?" Together → add. Apart → subtract.

Set Up the Right Operation 🔽

Decide whether each situation calls for adding or subtracting polynomials.

➕ Adding & Subtracting Polynomials

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) name polynomials and spot like terms, (2) add polynomials, (3) subtract by distributing the minus sign, and (4) apply all of this to real problems. Let's put it together.

Quick Reference

⚠️ Three things to never forget: (1) like terms need the same power; (2) only coefficients combine — exponents do not change; (3) when subtracting, the minus sign flips every term in the second group.

Mixed Practice 🧮

Simplify each expression and enter the coefficients of the result, in order from highest power to lowest.

1) $(6x^2 - 3x + 4) + (x^2 + 3x - 9)$   Enter the coefficient of $x^2$, then $x$, then the constant.

2) $(10x^2 + 7x) - (4x^2 + 7x - 5)$   Enter the coefficient of $x^2$, then $x$, then the constant.

When Terms Disappear

Did you notice the $x$-terms vanished in both problems above? That is completely normal — whenever like terms have opposite coefficients (like $7x$ and $-7x$), they add to $0$ and drop out of the answer.

💡 A missing term is not a mistake. $7x^2 - 5$ simply has no $x$-term — its coefficient is $0$. The next questions include more of these "disappearing term" cases, so stay alert.

Mixed Practice 🎯

Exit Quiz ✅

Answer all three to finish the lesson.